# Annual interest rate from different monthly interest rates

Let's says I have unequal monthly interest rates (1,5% in the first month, 1,7% in the second, -0,8% in the third...) is there a way I can calculate what was my interest rate in the semester or in the year? On all the formulas I could manage to find it is taken into consideration that monthly rates are always the same.

• It'd be easiest to handle this in a spreadsheet where you can copy down, or in a script that can loop over the changing rates, but an annualized rate would be calculated with: `APY = (1+r1)*(1+r2) *(1+r3) *(1+r4) *(1+r5) *(1+r6) *(1+r7) *(1+r8) *(1+r9) *(1+r10) *(1+r11) *(1+r12) – 1` where r1-r12 are the 12 different monthly rates. If you're talking about variable rate debt, then the above neglects compounding interest. – Hart CO Jan 16 '18 at 18:45
• @Hartco Any reason that comment shouldn't be an answer? – Rupert Morrish Jan 16 '18 at 21:16
• @RupertMorrish I suspected they need to incorporate compounding and I didn't want to fuss with showing what the copy-down formula would be in Excel. – Hart CO Jan 16 '18 at 21:50
• @HartCO Should be `1+(r1/12)` and so on, assuming the interest rates are quoted as annual rates, which they almost certainly are. – David Schwartz Jan 17 '18 at 0:30

You can use the geometric mean, e.g.

Check

``````quarterly return = (1 + 0.015)*(1 + 0.017)*(1 - 0.008) - 1 = 2.3997 %
``````

Using the geometric mean

``````gm = 1.007935842

quarterly return = gm^3 - 1 = 2.3997 %
``````

So the average monthly return is `gm - 1 = 0.7935842 %`

The effective annual interest rate would be `gm^12 - 1 = 9.94986 %` and the nominal annual rate compounded monthly would be `(gm - 1)*12 = 9.52301 %`.